A Boundary Crisis in High Dimensional Chaotic Systems

  • Ling Hong
  • Yingwu Zhang
  • Jun Jiang
Conference paper


A crisis is investigated in high dimensional chaotic systems by means of generalized cell mapping digraph (GCMD) method. The crisis happens when a hyperchaotic attractor collides with a chaotic saddle in its fractal boundary, and is called a hyperchaotic boundary crisis. In such a case, the hyperchaotic attractor together with its basin of attraction is suddenly destroyed as a control parameter passes through a critical value, leaving behind a hyperchaotic saddle in the place of the original hyperchaotic attractor in phase space after the crisis, namely, the hyperchaotic attractor is converted into an incremental portion of the hyperchaotic saddle after the collision. This hyperchaotic saddle is an invariant and nonattracting hyperchaotic set. In the hyperchaotic boundary crisis, the chaotic saddle in the boundary has a complicated pattern and plays an extremely important role. We also investigate the formation and evolution of the chaotic saddle in the fractal boundary, particularly concentrating on its discontinuous bifurcations (metamorphoses). We demonstrate that the saddle in the boundary undergoes an abrupt enlargement in its size by a collision between two saddles in basin interior and boundary.


Chaotic Attractor Fractal Boundary Positive Lyapunov Exponent Unstable Node Basin Interior 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This work is supported by the National Science Foundation of China under Grant Nos. 10772140 and 10872155 as well as the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.


  1. 1.
    Baier G, Klein M (1990) Maximum hyperchaos in generalized Henon map. Phys Lett A 151:281–284MathSciNetCrossRefGoogle Scholar
  2. 2.
    Baier G, Sahle S (1995) Design of hyperchaotic flows. Phys Rev E 51:R2712–R2714CrossRefGoogle Scholar
  3. 3.
    Rossler OE (1979) An equation for hyperchaos. Phys Lett A 71:155–157MathSciNetCrossRefGoogle Scholar
  4. 4.
    Matsumoto T, Chua LO, Kobayashi K (1986) Hyperchaos: laboratory experiment and numerical confirmation. IEEE Trans Circuits Syst CAS-33(11):1143–1147MathSciNetCrossRefGoogle Scholar
  5. 5.
    Kapitaniak T, Thylwe KE, Cohen I, Wjewoda J (1995) Chaos–hyperchaos transition. Chaos Solitons Fractals 5(10):2003–2011MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Reiterer P, Lainscsek C, Schurrer F (1998) A nine-dimensional lorenz system to study high-dimensional chaos. J Phys A 31:7121–7139CrossRefMATHGoogle Scholar
  7. 7.
    Kapitaniak T, Maistrenko Y, Popovych S (2000) Chaos–hyperchaos transition. Phys Rev E 62(2):1972–1976CrossRefGoogle Scholar
  8. 8.
    Kapitaniak T (2005) Chaos synchronization and hyperchaos. J Phys: Conf Ser 23:317–324MathSciNetCrossRefGoogle Scholar
  9. 9.
    Ott E, Sommerer JC (1994) Blowout bifurcations: the occurrence of riddled basins and on–off intermittency. Phys Lett A 188:39–47CrossRefGoogle Scholar
  10. 10.
    Kapitaniak T, Lai YC, Grebogi C (1999) Metamorphosis of chaotic saddle. Phys Lett A 259(6):445–450MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ditto WL, Rauseo S, Cawley R, Grebogi C (1989) Experimental observation of crisis-induced intermittency and its critical exponent. Phys Rev Lett 63:923–926CrossRefGoogle Scholar
  12. 12.
    Grebogi C, Ott E, Yorke JA (1982) Chaotic attractors in crisis. Phys Rev Lett 48:1507–1510MathSciNetCrossRefGoogle Scholar
  13. 13.
    Ott E (2002) Chaos in dynamical systems. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  14. 14.
    Hsu CS (1995) Global analysis of dynamical systems using posets and digraphs. Int J Bifurcat Chaos 5(4):1085–1118CrossRefMATHGoogle Scholar
  15. 15.
    Hong L, Xu JX (1999) Crises and chaotic transients studied by the generalized cell mapping digraph method. Phys Lett A 262:361–375MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hong L, Xu JX (2001) Discontinuous bifurcations of chaotic attractors in forced oscillators by generalized cell mapping digraph (GCMD) method. Int J Bifurcat Chaos 11:723–736MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Hong L, Sun JQ (2006) Codimension two bifurcations of nonlinear systems driven by fuzzy noise. Physica D: Nonlinear Phenom 213(2):181–189MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Xu W, He Q, Fang T, Rong H (2004) Stochastic bifurcation in Duffing system subject to harmonic excitation and in presence of random noise. Int J Non-Linear Mech 39:1473–1479MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kawakami H, Kobayashi O (1976) Computer experiment on chaotic solution. Bull Fac Eng Tokushima Univ 16:29–46Google Scholar
  20. 20.
    He DH, Xu JX, Chen YH (1999) A study on strange dynamics of a two dimensional map. Acta Physica Sin 48(9):1611–1617Google Scholar
  21. 21.
    He DH, Xu JX, Chen YH (2000) Study on strange hyper chaotic dynamics of kawakami map. Acta Mechanica Sin 32(6):750–754MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.MOE Key Lab for Strength and Vibration, School of AerospaceXi’an Jiaotong UniversityXi’anPeople’s Republic of China

Personalised recommendations